Supporting Information

Mass Action at the Single-Molecule Level

Min Ju Shon and Adam E. Cohen*

*cohen@chemistry.harvard.edu Departments of Chemistry and Chemical Biology and of Physics, Harvard University, Cambridge, Massachusetts 02138, United States

Table of Contents

Part I. Construction of the Dimple Machine

Figure S1. Nanofabrication of dimple arrays

Figure S2. Microfabrication of the lid with microposts

Figure S3. Lid-dimple assembly

Part II. Methods for the Dimple Machine

Section S1. Experimental setup and procedures

Section S2. Optimization

Table S1. Composition of the trapping buffer

Table S2. Composition of the deoxygenation buffer

Part III. Verification of the Dimple Machine

Figure S4. Suppression of photobleaching

Figure S5. Counting of fluorescent dyes in a dimple

Figure S6. Reproducible filling of dimples

Part IV. Measurement of interaction using red-green correlation

Table S3. DNA sequences

Figure S7. Measurement of Kd from correlated occupancy

Table S4. Fitting parameters for a correlated bivariate Poisson model

Section S3. Theory of correlated bivariate Poisson distribution

Part V. DNA hybridization in dimples

Section S4. Bulk measurement of Kd

Figure S8. Bulk measurement of Kd

Section S5. Theory of multi-state Markov process

S1  

Part I. Construction of the Dimple Machine

Figure S1. Nanofabrication of dimple arrays

(A) Scheme for the nanofabrication. (a) A 1" square fused silica coverslip (Esco, R425025 S1-UV) was cleaned

in piranha solution (3:1 sulfuric acid : hydrogen peroxide) at 70 °C for 30 min, followed by rinsing in water. The

substrate was then spin-coated with adhesion promoter (ShinEtsu, MicroSi MicroPrime HP Primer, at 1,000 rpm),

electron-beam lithography resist poly(methyl methacrylate) (PMMA, MicroChem, 950 A4, at 3,000 rpm), and con-

ducting polymer (electron-discharge layer, Showa Denko, Espacer300, at 1,000 rpm). (b) Electron-beam lithogra-

phy was carried out by JEOL JSM-7000F at an accelerating voltage of 30 kV and an electron dose in the range of

700–1,200 C/cm2. Typically 40 dimple arrays were written on one substrate during the exposure, and each array

(120 m × 120 m) contained 900 circular dimples of varying sizes, arranged in a square lattice with spacing 4 m.

(c) The processed substrate was (1) washed in water to dissolve the conducting polymer layer, (2) developed in 1:3

methyl isobutyl ketone : isopropyl alcohol (IPA) for 2 min, and (3) washed in IPA. (d) The developed substrate was

briefly cleaned by oxygen plasma (Technics Micro-stripper, 30 mTorr of oxygen for 30 s) and the exposed fused

silica was etched in a reactive ion etching system (STS-ICP-RIE) by fluorine-based recipe using PMMA as an etch

mask. (e) PMMA after etching was completely removed by dissolving in acetone and in 2 % Hellmanex solution in

S2  

water at 70 °C for 30 min. (f) The nanofabricated fused silica was annealed in a furnace at 900 °C for 5 h to remove

the autofluorescence from the fused silica induced by the e-beam lithography (Section S2).

(B) Clean surface of a fused silica coverslip. (top) Atomic microscope (AFM, Asylum Research, MFP-3D) image

of the surface of a fused silica coverslip after cleaned in piranha solution. Root-mean-square surface roughness was

0.4 nm. (bottom) Section of the image along the red line in (top).

(C) Nanofabricated dimples on a fused silica coverslip. (top) Images of the dimples with nominal radii of (a) 150,

(b) 250, (c) 350, (d) 450, (e) 550, (f) 650 nm. (bottom) Section of the scan along the red dashed line in the (top)

image. The final depth of etching was measured to be 190–200 nm for all dimples.

(D) Autofluorescence of the fused silica. Surface of a fused silica coverslip imaged on a fluorescence microscope

exciting at 633 nm. All images are displayed with the same brightness and contrast. (top) Initial autofluorescence

was sufficiently low to allow detection of single-molecules. (middle) Electron-beam lithography led to an increase

in autofluorescence by > 1000-fold, in an electron-dose-dependent manner (Section S2). (bottom) Baking the fin-

ished fused silica wafer in a furnace at 900 °C for 5 h completely eliminated the electron beam-induced fluorescence

(Section S2), without distorting the nanofabricated dimples.

S3  

Figure S2. Microfabrication of the lid with microposts

(A) Scheme for the Microfabrication of the lid. (a) A 3" silicon wafer was cleaned by sonicating in acetone, IPA,

and methanol, successively, for 5 min each, rinsed in water, and dried on a hotplate. The cleaned wafer was coated

with a negative photoresist SU-8 (MicroChem, SU-8 3025) and baked on a hotplate at 95 °C for 15 min. (b) The

cured resist was exposed through a transparency mask to UV for 15 min. The substrate was baked on a hotplate at

65–95 °C for 5 min. (c) The resist was developed in SU-8 developer (MicroChem) for 10 min. The wafer was thor-

oughly rinsed with IPA and hard-baked at 200 °C for 30 min. (d) The fabricated SU-8 microstructures served as a

master for the fabrication of the PDMS lid. Two-part mixture of PDMS (Dow Corning, Sylgard 184, 10:1 base :

curing agent) was poured over the master and topped with a bare silicon wafer to control the thickness of the PDMS

membrane to 500 m. The assembly was then cured at 80 °C for 5 h. (e) The top wafer was removed carefully and

the cured PDMS membrane was cut to 20 mm × 20 mm and peeled off the master.

S4  

(B) Microfabriated SU-8 master for the lid. (top) Photo of the surface and (bottom) surface scan with a contact

stylus profiler (KLA-Tencor P-16+) along the red dashed line in the (top) image. The patterns contained squares (a)

150 m on a 200 m grid or (b) 300 m on a 400 m grid; or rectangles (c) 500 m × 150 m on a 600 m ×

200 m grid, and (d) 1000 m × 300 m on a 1200 m × 400 m grid. The size and spacing of the posts was not

critical; for each device an array of posts was selected with dimensions that facilitated sealing of complete arrays of

dimples. The height of the posts was 28 m in all structures.

S5  

Figure S3. Lid-dimple assembly

(A) Scheme for the lid-dimple assembly. (a) Three layers of PDMS were prepared: (1) a deoxygenation chamber

with ϕ = 5 mm hole in a slab (3-mm-thick), (2) the microfabricated lid membrane with microposts (500-m-thick)

(Figure S2), and (3) a spacer with a hexagonal sample channel cut out with a knife (500-m-thick). Surfaces of the-

se layers were activated by oxygen plasma for 1 min (SPI, Plasma-Prep II) and then irreversibly bonded. (b) Two

ports for sample injection were punched into the PDMS lid complex using a blunt needle (18G). The microposts on

the lid were aligned to the dimple arrays by assembly under a dissecting microscope with a custom three-axis aligner

(Thorlabs, DT12). The reversible bonding between PDMS and fused silica was strong enough to remain stable

throughout the experiment and allowed disassembly, cleaning, and reuse without sacrificing the lid or the substrate.

(B) Cartoon of the sample channel. The lid is shown when (left) sealing and (right) refreshing the dimples.

(C) Microscope images of the PDMS lid assembly. (left) A full view of the sample channel from the top. (mid-

dle) Micropost arrays placed on the dimple substrate. (right) A close view of the dimple arrays aligned under the

rectangular microposts.

S6  

Part II. Methods for the Dimple Machine

Section S1. Experimental setup and procedures

Optical setup. An alternating laser excitation (ALEX) setup equipped with a dual-view EM-CCD (Andor iXon,

DU-897E-CS0-#BV, 512 × 512 with 16 m pixels) was constructed (Figure 1) by adapting protocols described

elsewhere.S1,S2 Briefly, two laser beams (638 and 532 nm) were combined using a dichroic mirror, and the intensity

of each was modulated by an acousto-optical tunable filter (Gooch & Housego, PCAOM 48058-2.5-.55-5W). The

microscope contained a dual-band excitation filter (Chroma, z532/635rpc). The molecules were illuminated in epif-

luorescence mode with a 60 X oil-immersion objective (Olympus, 1-U2B616, N.A. 1.45) mounted on an inverted

epifluorescence microscope (Olympus, IX71). Typical illumination intensities were 100 W/cm2. The fluorescence

image exiting the microscope was cropped to half with a slit, and then separated into two channels (red and green)

by a dichroic mirror. Each channel was imaged on one half of the CCD detector.

Sample preparation. Three sets of fluorescently labeled single-stranded DNA (ssDNA) oligos were purchased

from IDT: a non-complementary pair, a doubly labeled hairpin, and a hybridizing pair (Table S3).

Preparation for trapping. Prior to trapping, the sample channel was incubated with 1 % polyvinylpyrrolidone

(PVP, Sigma-Aldrich) for 2 h to minimize adsorption of molecules on the fused silica. The sample channel was then

washed with 0.1 % PVP multiple times, and briefly degassed in a desiccator. The experimental sample (40 L) was

then injected into the sample channel. All trapping experiments were carried out in the “trapping buffer” containing

25 mM HEPES at pH 8.0 with 150 mM sodium chloride and with low concentrations of additives to minimize sur-

face effect and improve photostability (Table S1). The deoxygenation reservoir on the top of the PDMS lid was

filled with 200 L of deoxygenation buffer (Table S2) to further deoxygenate the sample (Figure S4). Polyethylene

tubing (BD, PE-90) was inserted into the ports of sample channel, and connected to house vacuum via a solenoid

valve (Warner, VC-8). The device was placed on a microscope stage and then enclosed in a plastic box containing

wet towels to maintain humidity. The box was purged with a constant flow (< 2 scfh) of humidified nitrogen

throughout the measurement.

Trapping and measurement. The scheme for pneumatically actuating the PDMS lid (Figure 1) was adapted from

Ref. S3. Briefly, the lid was actuated by applying vacuum to both ports of the sample chamber. Atmospheric pres-

sure on the top of the lid then caused the lid to bow down until it contacted the silica coverslip, sealing the dimples.

S7  

We found that it was typically necessary to wait ~5 min between actuation of the vacuum and imaging for the lid to

achieve a hermetic seal with the fused silica. Each cycle consisted of 90 s of data acquisition at an exposure time of

50 ms/frame. Red and green laser excitations were applied on alternate frames. The EM-CCD gain was 300. To

refresh the dimples, the vacuum was released and the lid returned to its raised position. The lid actuation, alternat-

ing laser excitation, and camera exposures were synchronized by a Data Acquisition board (National Instruments,

PCIe-6323) run by custom software (LabView, National Instruments).

Device storage. After experiments the Dimple Machine was washed by flowing 0.1 % PVP and 0.1 % Tween-20

(Sigma-Aldrich) through the sample channel. The washed device was stored in 4 °C with 0.1 % PVP and 0.02 %

sodium azide in the sample channel to inhibit the growth of bacteria. It was not necessary to condition the device

with 1 % PVP again for the next experiment. If needed, the dimple array substrate was detached from the PDMS lid,

cleaned in piranha solution, and bonded to the lid again.

Data analysis. Data was analyzed with custom software written in MATLAB (R2011a, MathWorks). Briefly, three

sets of fluorescence time traces were extracted: (a) D(t): green emission under green excitation; (b) A(t): red emis-

sion under green excitation; and (c) R(t): red emission under red excitation. The fluorescence intensity from each

dimple was extracted from the 7×7 pixels (typical) centered on the dimple, and the background fluorescence esti-

mated from neighboring pixels was subtracted from the dimple intensity. To estimate the occupancy by red mole-

cules, we looked at R(t); to estimate the occupancy by green molecules, we looked at G(t) = D(t) + A(t). The occu-

pancy was measured by counting the number of fluorescent dyes as described in Figure S5. A 2D histogram of red

and green fluorescence (R-G histogram) was constructed in a similar way to Figure S5, but with two traces R(t) and

G(t). When analyzing FRET, a 2D histogram of acceptor and donor fluorescence (A-D histogram) was constructed

similarly using D(t) and A(t). Data traces in which we observed a photobleaching event were excluded from kinetic

or thermodynamic analysis.S1

S8  

Section S2. Optimization

Background autofluorescence from fused silica. Minimizing background fluorescence in single-molecule meas-

urements is crucial to enhance signal-to-noise ratio. Fused silica shows very low autofluorescence. However, we

observed a high level of autofluorescence (especially under red excitation) in the regions of the sample exposed to

the electron beam during electron-beam lithography. While this phenomenon was documented previously,S4 we did

not find any remedies in the literature. We discovered that by baking the substrate in furnace at 900 °C for 5 h the

autofluorescence decreased to the initial low level.

Reproducibility of trapping. To achieve reproducible trapping, we found that it was necessary to lower the PDMS

lid at the same rate, and with the same force, every time. To achieve this reproducibility, actuation of the lid was

controlled by a digital vacuum valve with precise timing to open and close (Section S1). This feature ensured re-

producible repetition of the trapping, producing a large dataset without artifact (Figure S6).

Nonspecific adsorption of molecules on the fused silica and PDMS surface. Biomolecules such as DNA and

proteins adsorb onto many surfaces, especially on hydrophobic ones. This nonspecific sticking creates background

fluorescence that interferes with observations of single molecules. Polyvinylpyrrolidone (PVP) generates a physi-

cally adsorbed barrier that prevents nonspecific adsorption of DNA on the fused silica and PDMS.S6,S7 Incubation

with PVP solution for 2 h prior to the first experiment, suppressed sticking of DNA oligos on fused silica surface

effectively. We also mixed the sample with PVP (0.1%) and Tween-20 (0.1%) to minimize sticking of molecules on

the fused silica and PDMS surface, respectively (Figure S6C).

Suppression of photobleaching. Photobleaching can be suppressed by enzymatic deoxygenation of the solution.

We first used protocatechuic acid (PCA) / protocatechuate-3,4-dioxygenase (PCD) to deoxygenate the sample en-

zymatically.S5 However, this approach was not effective because (1) The PCD enzyme was typically at a concentra-

tion so low that many dimples contained zero PCD molecules; and (2) Oxygen has high solubility in and permeabil-

ity through PDMS, so the PDMS lid acted as a large reservoir of oxygen, easily overwhelming the capacity of the

oxygen scavengers in the dimples. We improved deoxygenation by combining three approaches (Figure S4A).

(1) Enzymatic oxygen scavenging in the microchannels. We designed a PDMS lid with microposts. When the lid

was closed, the inverted posts on the lid sealed the dimple array, creating microchannels around the array. The

S9  

channels then irrigated bulk solution near to the dimple array (~ 100 m), providing a large sink for oxygen close to

the dimples.

(2) Remote deoxygenation from across the lid membrane. We exploited the oxygen permeability of PDMS to de-

oxygenate the dimples from the top side of the PDMS lid. The reservoir on top of the PDMS lid was loaded with

deoxygenation buffer (Table S2). This reservoir generated a reversed oxygen concentration gradient spanning the

lid membrane, causing oxygen to diffuse out of the dimples and into the PDMS lid.

(3) Operation in a nitrogen atmosphere. A constant flow of house N2 was blown into a humidified box covering the

device. This physical exclusion of oxygen from the sample chamber prolonged the lifetime of the oxygen scavenger

reservoir, facilitating long-term measurements.

In addition to PCA/PCD, we added methyl viologen (1 mM) and ascorbic acid (1 mM) as a redox system quenching

the triplet states of fluorescent dyes, and thereby suppressing photoblinking.S8

Leakage of molecules out of the dimples. Reliable sealing of the dimples is a prerequisite of trapping and accurate

measurement of occupancy. The surface roughness of the substrate must be smaller than the length-scale of the

molecules to be trapped, typically ~1 nm. Clean fused silica has surface roughness below 1 nm (Figure S1B).

However, plasma etching or exposure to extremes of pH gradually roughened the surface, necessitating care in

cleaning. A thin aqueous film often persisted between the fused silica and the lid immediately after closing the lid.

This film permitted molecules to escape from the dimples. To facilitate the draining of this film, we introduced the

post topography into the PDMS lid, thereby minimizing the distance the water had to flow to reach a deep channel.

After introducing the posts, we found that the lid sealed hermetically after ~5 min.

Diffusion of water into the PDMS lid leading to drying of the dimples. PDMS absorbs water to a small extent.

Each sealed dimple contained < 1 fL of water, which easily was absorbed into the PDMS lid. The deoxygenation

buffer on top of the lid also saturated the lid with water, preventing additional absorption of water from the dimples.

The device was run inside a humidified chamber to prevent drying of the deoxygenation buffer. The sealed dimples

remained hydrated until the deoxygenation buffer completely dried out, usually after more than 1 day.

Avoidance of air bubbles in the sample channel. Application of vacuum to the sample channel induced formation

of micro-bubbles in the channel. To avoid bubbles, (1) the sample solution was injected with a micropipette with

minimal speed of loading, and (2) the device was degassed for a few minutes in a vacuum desiccator prior to use.

S10  

Table S1. Composition of the trapping buffer

Material Concentration Role

HEPES 25 mM Buffer at pH 8.0

NaCl 150 mM Ionic strength

Polyvinylpyrrolidone (Avg. MW 10 kDa) 0.1 % Prevents sticking on fused silica

Tween 20 0.1 % Prevents sticking on PDMS

Methyl viologen dichloride 1 mM Triplet-state quencher (oxidant)

Sodium L-ascorbate 1 mM Triplet-state quencher (reductant)

Protocatechuic acid 10 mM Enzymatic deoxygenation (substrate)

Protocatechuate 3,4-dioxygenase 0.1 M Enzymatic deoxygenation (enzyme)

* All materials in Table S1 and S2 were purchased from Sigma-Aldrich.

Table S2. Composition of the deoxygenation buffer

Material Concentration Role

HEPES 25 mM Buffer at pH 8.0

Tween 20 0.1 % Prevents sticking on PDMS

Protocatechuic acid 10 mM Enzymatic deoxygenation (substrate)

Protocatechuate 3,4-dioxygenase 0.1 M Enzymatic deoxygenation (enzyme)

* All materials in Table S1 and S2 were purchased from Sigma-Aldrich.

S11  

Part III. Verification of the Dimple Machine

Figure S4. Suppression of photobleaching

(A) Scheme for the suppression of photobleaching. The dimples were deoxygenated by three methods: (1) En-

zymatic oxygen scavenging in the microchannels; (2) Remote deoxygenation from across the lid membrane; and (3)

Nitrogen atmosphere surrounding the device (Section S2). Note that water molecules can also diffuse through

PDMS, indicated by the white dashed arrows (Section S2).

(B) Decay of fluorescence due to photobleaching. The fluorescence is averaged over 100 dimples and 60 cycles.

Less than 3 % of fluorescence photobleached in 1 min.

(C) Prolonged observation of single molecules. Fluorescence intensity traces showing single-step photobleaching

for (a) 1, (b) 2, (c) 3, and (d) 4 fluorescent dyes in a single dimple. In these experiments the lid was kept sealed

throughout.

S12  

Figure S5. Counting of fluorescent dyes in a dimple

(A) Fluorescence image of a dimple array averaged over time. The yellow circle marks the dimple whose time

trace is plotted in (B).

(B) Fluorescence time trace of a dimple. The data is concatenated over 60 trapping cycles. Cycles lasted 1 min

and were spaced by 5 min. The intensity histogram of the trace is attached on the right. The unit fluorescence in-

tensity from a single dye was obtained from the average spacing of peaks in the histogram.

(C) A map unit fluorescence of a dimple array. The intensity of each pixel represents the unit fluorescence for

each dimple shown in (A). The total fluorescence from each dimple was normalized by the corresponding unit fluo-

rescence to measure occupancy.

(D) Histogram of normalized fluorescence. The graph is for 2891 time traces recorded from across the array.

S13  

Figure S6. Reproducible filling of dimples

The quantity N(k, x) represents the occupancy of the k-th dimple in the x-th cycle of trapping.

(A) The occupancy distribution for individual dimples averaged over cycles. The dashed line indicates the av-

erage occupancy over all dimples. This distribution shows that there were not significant variations in dimple vol-

ume.

(B) The occupancy distribution for individual cycles averaged over dimples. The dashed line indicates the aver-

age occupancy over all cycles. This distribution shows that there were not significant variations in mean dimple

occupancy from cycle to cycle.

(C) Correlation in the single-dimple occupancy over consecutive cycles. The distribution of correlations (for 60

cycles) is centered on zero, indicating that the loading of the dimples was uncorrelated between successive cycles.

This result confirms complete interchange of dimple contents during each refresh cycle.

S14  

Part IV. Measurement of interaction using red-green correlation

Table S3. DNA sequences

Linkage Sequence Length (bp)

R: 5'-/Cy5/-TCT AGT CTC CCC AAA AAA AAA AAA AAA AAA-3' 30 None

G: 5'-ATG TCT GAC GCT GGC ATT CGC ATC AAA GGA-/Cy3/-3' 30

Covalent

RG: 5'-GCA ATT TAT TAA TTA TAT ATT TTA TTT AAT ATA ATT/Alexa647/ CCT GGA TCC GCA AGC TGG CGT TTA GTG ATA TCC CGT ATA AGT CTG TAG TGA ATT CTC C/Cy3B/TA TTA TAT TAA ATA AAA TAT ATA ATT AAT AAA TTG C-3'

130

R: 5'-/Cy5/-TCT AGT CTC CCC AAA AAA AAA AAA AAA AAA-3' 30 Hybridization

G: 5'-/Cy3/-AGA CTA GAC CCC AAA AAA AAA AAA AAA AAA-3' 30

* All DNA samples were ordered form IDT. The ssDNA oligo “R” of the hybridization pair was used in the proof-

of-principle occupancy measurement in Figure 2 and Figure S4–S6.

S15  

Figure S7. Measurement of Kd from correlated occupancy

(left) Non-complementary, (center) covalently linked, and (right) hybridizing ssDNA pairs.

(A) R-G histogram of the unfiltered raw fluorescence. The R-G histograms shown in the main text were obtained

from the raw fluorescence traces by application of a median filter running over 13 frames.

(B) R-G histogram of the rounded fluorescence. The mean normalized fluorescence from each dimple in each

cycle was rounded to the nearest integer value, to provide an estimate of occupancy.

(C) Correlated bivariate Poisson fit to the R-G histograms in (B). Fitting parameters are given in Table S4.

S16  

Table S4. Fitting parameters for a correlated bivariate Poisson model

Linkage None Covalent Hybridization

R 2.21 ± 0.11 0.31 ± 0.04 1.39 ± 0.13

G 3.43 ± 0.12 0.96 ± 0.05 1.09 ± 0.14

RG 0.39 ± 0.10 2.50 ± 0.07 1.07 ± 0.13

RG 0.12 ± 0.03 0.80 ± 0.06 0.46 ± 0.07

Kd (nM) 900 ± 200 1.10 ± 0.15 63 ± 12

* The dimple occupancy by two spectrally distinguishable interacting species (R and G in the main text) was mod-

eled as a correlated bivariate Poisson distribution (Section S3).

S17  

Section S3. Theory of correlated bivariate Poisson distribution

When a mixture of R, G, and RG is trapped in dimples, each species assorts independently, following its own

Poisson distribution:

!

)()(

RR

R N

eNf

N

, !

)()(

GG

G N

eNf

N

, and !

)()(

RGRG

RG N

eNf

N

(1)

where R, G, and RG are the average occupancy of dimples by R, G, and RG, respectively. Consider a dimple with

a joint occupancy, . If there are NRG molecules of RG in this dimple, the occupancy by monomeric R

and G are and , respectively. The probability of the joint distribution is

),( (T)G

(T)R NN

RG(T)R NR NN RG

(T)GG NNN

(2)

maxRG

RG 0RGRGRG

(T)GGRG

(T)RR

(T)G

(T)R )()()(),(P

N

N

NfNNfNNfNN

Plugging in Eq. 1 into Eq. 2 results in the following expression:

RG

GR(T)R

(T)G

(T)G

RG

GRGR

)((T)G

(T)R ,1,)()(),(P

(T)G

(T)G

(T)RRGGR

NNNUeNN

N

NN (3)

where U represents the Tricomi confluent hypergeometric function:

0

11 )1()(

1),,( dttte

azbaU abazt . (4)

Eq. 3 serves as a model for the joint distribution with three fitting parameters R, G, and RG. After fitting the ex-

perimental R-G histogram to Eq. 3, the correlation coefficient RG between R and G is obtained by

))(( RGGRGR

RGRG

. (5)

One can infer the equilibrium concentration of species in bulk from the average occupancy of dimples (Figure S7,

Table S4). The dissociation constant is:

)/(

))/())(/((

RG][

G][R][

dARG

dAGdAR

eq

eqeqd VN

VNVNK

. (6)

S18  

Part V. DNA hybridization in dimples

Section S4. Bulk measurement of Kd

Experimental procedure. Equimolar mixtures of two hybridizing ssDNA oligos (Table S3) were prepared at vari-

ous concentrations from 2 nM to 2 in HEPES buffer at pH 8.0 with 150 mM NaCl. The fluorescence emission

spectra of the mixture were measured in a fluorimeter (Cary Eclipse, Agilent) exciting the sample at 550 nm, col-

lecting the emission from 550 nm to 750 nm (Figure S8A). The emission intensity at 670 nm was selected as a val-

ue representing the intensity of FRET acceptor emission and plotted against concentrations (Figure S8B).

Analysis: Fraction of dimer in a binary mixture. When R and G associate in a mixture, they exist in equilibrium

with the bound form, RG:

R + G ⇄ RG (1)

with the relative fraction of each species determined by the dissociation constant Kd:

eq

eq0eq0

eq

eqeqd ]RG[

)]RG[]G)([]RG[]R([

]RG[

]G[]R[ K (2)

where [R]eq, [G]eq, and [RG]eq are the equilibrium concentrations of R, G, and RG, respectively, and [R]0 and [G]0

are the initial concentrations of R and G, respectively. Solving Eq. 2 for [RG]eq as a function of [R]0, [G]0, and Kd

gives

002

d00d00eq G][R][4)G][R]([G][R][2

1RG][ KK . (3)

In our study, the FRET efficiency is low when R is unbound, and high when bound to G. If the acceptor emission

intensity is Ilow in R and Ihigh in RG, the average acceptor intensity IA is

)(R][

RG][

R][

RG][

R][

R][lowhigh

0

eqlowhigh

0

eqlow

0

eq IIIIII A

(4)

Substituting Eq. 3 into Eq. 4 gives a model for the acceptor emission. This model was fit to the experimental curve

to obtain Kd (Figure S8B).

S19  

Figure S8.  Bulk measurement of Kd 

 

See Section S4 for the methods of experiment and analysis.

(A) Fluorescence emission spectra of the hybridizing ssDNA pair mixtures. The spectra are measured in a

fluorospectrometer at the excitation wavelength of 550 nm at various initial concentrations (legend). The dashed

line (emission at 670 nm) indicates the wavelength at which the FRET acceptor emission is taken.

(B) FRET acceptor emission intensity as a function of initial concentration of DNA, [R]0. The Kd extracted

from the curve was 57 ± 12 nM. Exp: Experimental data; Fit: Fitted curve.

S20  

Section S5. Theory of multi-state Markov process

Steady-state distributions

Consider a dimple with a joint occupancy in which R and G associate and dissociate with rate

constants kon and koff, respectively. Assuming that the reaction is Markovian, the number of dimers at time t, NRG (t)

can be considered as a state variable whose path is determined by the kinetic matrix M with the element mij repre-

senting the transition rate from state j to state i:

),( (T)G

(T)R NN

Δt

jtNiΔttNm ij

Δtij

])()(P[lim RGRG

0 (1) ),,1,0,( max

RGNji

where is the maximum number of RG given by . Note that to avoid confusion, the matrix

indices i and j start from zero. Only certain elements of the rate matrix are non-zero. These are:

maxRGN ),min( (T)

G(T)R NN

dA

G

dA

Ronon)1( R][G][

VN

N

VN

Nkkm ii

dA

(T)G

dA

(T)R

on

)()(

VN

iN

VN

iNk

, (2) )1,,1,0( max

RG Ni

dA

RGoffoff)1( RG][

VN

Nkkm ii

dA

off

1

VN

ik

, (3) )1,,1,0( max

RG Ni

, (4) iiiiij

jiii mmmm )1()1(

where NA is Avogadro’s number, and Vd is the volume of the dimple. All other elements of M are zero.

To calculate the steady state probability distribution, we apply the condition of detailed balance between state j

and j+1:

. (5) jjjjjj mm PP )1(1)1(

Applying this condition recursively from P0 yields:

0

1

0 )1(

)1( PP

i

j jj

jji m

m. (6) ),,2,1( RGNi max

S21  

Substituting Eqs. 2 and 3 into Eq. 6 gives an expression for the steady-state distributions relative to P0:

0offdA

on(T)G

(T)R

(T)G

(T)R P

)!()!(!

!!P

i

i kVN

k

iNiNi

NN

. (7)

Applying the condition yields: 1P i

jj

ii P

PP

. (8)

Eq. 10 can also be derived from equilibrium statistical mechanics as , where gi is the de-

generacy of state i and iG is the free energy to form i molecules of product.

TkGig

gB

ii /expP

P

00

An important feature of Eq. 7 is that the steady-state distribution of dimer depends not only on the reaction equilib-

rium (Kd = koff /kon), but also on the dimple volume and on the stochastically determined occupancy.

Autocorrelations

The second-order autocorrelation, C(2) () of the state variable i = NRG (t) is predicted by summing the condi-

tional probabilities of NRG (t) for the two time points separated by :

2

RG0

RG0

RGRG)2(

maxRG

maxRG

])0(P[])0()(P[)( NiNiiNjNjCN

i

N

j

(9)

(10) 2

eqeq )(1NPNP1N M e

where N is a diagonal matrix with Nii = i, and 1 is a row vector of ones.)1()1( maxRG

maxRG NN

S9

For the pairs with (i.e., two-state process), Eq. 10 reduces to a simple formula ),( (T)

G(T)R NN 1max

RG N

)(

2offon

offon)2( offon

)()( rre

rr

rrC

(11)

where ron and roff are the on- and off-rates of the reaction, respectively. Eq. 2 and 3 gives

)( dA

(T)G

(T)Ron

on VN

NNkr and . (12) offoff kr

S22  

The condition implies that at least one of or equals 1; however one of these quantities can be

greater than 1. The quantity NRG (t) was proportional to the acceptor fluorescence, so we calculated the autocorrela-

tion :

1maxRG N (T)

RN (T)GN

)()2( AAC

)()()()2( tAtACAA (13)

where δ represents the difference from the average. The autocorrelation was then fit to a model (Eq. 11) to extract

kon and koff (Figure 6b).

We calculated the autocorrelations for higher occupancies numerically by Eq. 9, keeping the kon and koff esti-

mates from the two-state autocorrelations. The calculated curves agreed well to the measurement for all high-

occupancy correlations, confirming that the rate constants are not dependent on occupancy.

S23  

S24  

References

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